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| {{DISPLAYTITLE:''p''-derivation}}
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| {{Merge to|Arithmetic_derivative|discuss=Talk:Arithmetic_derivative|date=September 2013}}
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| In [[mathematics]], more specifically [[differential algebra]], a '''''p''-derivation''' (for ''p'' a prime number) on a [[Ring (mathematics)|ring]] ''R'', is a mapping from ''R'' to ''R'' that satisfies certain conditions outlined directly below. The notion of a '''''p''-derivation''' is related to that of a [[Differential algebra|derivation]] in differential algebra.
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| ==Definition==
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| Let ''p'' be a prime number. A '''''p''-derivation''' or Buium derivative on a ring <math> R </math> is a map of sets <math> \delta:R\to R </math> that satisfies the following "[[product rule]]":
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| :<math> \delta_p(ab) = \delta_p (a)b^p + a^p\delta_p (b) + p\delta_p (a)\delta_p (b) </math>
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| and "sum rule":
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| :<math> \delta_p(a+b) = \delta_p (a) + \delta_p(b) + \frac{a^p +b^p - (a+b)^p }{p} </math>.
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| as well as
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| :<math> \delta_p(1) =0 </math>.
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| Note that in the "sum rule" we are not really dividing by ''p'', since all the relevant [[binomial coefficients]] in the numerator are divisible by ''p'', so this definition applies in the case when <math> R </math> has ''p''-[[Torsion (algebra)|torsion]].
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| ==Relation to Frobenius Endomorphisms==
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| A map <math> \sigma: R\to R </math> is a lift of the [[Frobenius endomorphism]] provided <math> \sigma(x) = x^p \mod pR </math>. An example such lift could come from the [[Artin map]].
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| If <math> (R,\delta) </math> is a ring with a ''p''-derivation, then the map
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| <math> \sigma(x) := x^p + p\delta(x) </math> defines a ring endomorphism which is a lift of the frobenius endomorphism. When the ring ''R'' is ''p''-torsion free the correspondence is a bijection.
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| ==Examples==
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| * For <math> R = \mathbb Z </math> the unique ''p''-derivation is the map
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| :<math> \delta(x) = \frac{x-x^p}{p}. </math>
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| The quotient is well-defined because of [[Fermat's Little Theorem]].
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| * If ''R'' is any ''p''-torsion free ring and <math>\sigma:R \to R</math> is a lift of the Frobenius endomorphism then
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| :<math> \delta(x) = \frac{\sigma(x)-x^p}{p} </math>
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| defines a ''p''-derivation.
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| ==See also==
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| *[[Derivation (abstract algebra)|Derivation]]
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| *[[Fermat Quotient]]
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| ==References==
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| * {{Citation|first=Alex|last=Buium|title=Arithmetic Differential Equations|year=1989|publisher=Springer-Verlag|isbn=0-8218-3862-8|series=Mathematical Surveys and Monographs}}.
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| ==External links==
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| *[http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.dmj/1077245037&page=record Project Euclid]
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| [[Category:Differential algebra]]
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I like Videophilia (Home theater).
I also try to learn English in my free time.
Look at my blog post Transfering to mountain bike sizing.